Inserting (REF ) (using \(N=1\) ) into (REF ) and ignoring all the terms in \(\Gamma _s^{\pm 2}\) , we can write the continuity equation as a function of the different modulation components. In other formulations, such as non-linear PSE (or even SPLSA [1]}), the explicit dependence on a single wavenumber \(\alpha _s\)  allows us to write three different equations from the equation above: one dependent on \(\Gamma _s^0=1\) , one on \(\Gamma _s\)  and another one on \(\Gamma _s^*=\Gamma _s^{-1}\) . The same argument is used here, by considering the collection to be performed at each streamwise station (since the shock-cell wavenumber is slowly varying, this approximation should still be representative of the phenomenon). Using that assumption, we can separate the evolution of disturbances related to the different powers of \(\Gamma _s\) . Namely, for \(\Gamma _s^0\)  we can write
\(\begin{split}\left[ -\mathrm {i}\omega \hat{\nu }_{0} + \bar{u}_{0} \left( \frac{\partial \hat{\nu }_{0}}{\partial x} + \mathrm {i}\alpha \hat{\nu }_{0} \right) + \hat{u}_{0} \frac{\partial \bar{\nu }_0}{\partial x} + \hat{v}_{0} \frac{\partial \bar{\nu }_{0}}{\partial r}+ \right. \\\left. - \hat{\nu }_{0} \frac{\partial \bar{u}_0}{\partial x} - \bar{\nu }_{0}\left( \frac{\partial \hat{u}_{0}}{\partial x} + \mathrm {i}\alpha \hat{u}_{0} \right) - \bar{\nu }_{0}\left( \frac{\partial }{\partial r} + \frac{1}{r}\right) \hat{v}_{0} - \bar{\nu }_{0}\frac{\mathrm {i}m}{r} \hat{w}_{0} \right] + \\+\left[ u_s^* \left( \frac{\partial \hat{\nu }_{+1}}{\partial x} + \mathrm {i}(\alpha +\alpha _s) \hat{\nu }_{+1} \right) + \hat{u}_{+1}\left( \frac{\partial \nu _s^*}{\partial x} - \mathrm {i}\alpha _s \nu _s^* \right) + \hat{v}_{+1} \frac{\partial \nu _s^*}{\partial r}+ \right. \\\left. - \hat{\nu }_{+1} \left( \frac{\partial u_s^*}{\partial x} - \mathrm {i}\alpha _s u_s^* \right) - \nu _{s}^*\left( \frac{\partial \hat{u}_{+1}}{\partial x} + \mathrm {i}(\alpha +\alpha _s )\hat{u}_{+1} \right) - \nu _{s}^*\left( \frac{\partial }{\partial r} + \frac{1}{r}\right) \hat{v}_{+1} - \nu _s^*\frac{\mathrm {i}m}{r} \hat{w}_{+1} \right] + \\+\left[ u_s \left( \frac{\partial \hat{\nu }_{-1}}{\partial x} + \mathrm {i}(\alpha -\alpha _s) \hat{\nu }_{-1} \right) + \hat{u}_{-1}\left( \frac{\partial \nu _s}{\partial x} + \mathrm {i}\alpha _s \nu _s \right) + \hat{v}_{-1} \frac{\partial \nu _s}{\partial r}+ \right. \\\left. - \hat{\nu }_{-1} \left( \frac{\partial u_s}{\partial x} + \mathrm {i}\alpha _s u_s \right) - \nu _{s}\left( \frac{\partial \hat{u}_{-1}}{\partial x} + \mathrm {i}(\alpha -\alpha _s )\hat{u}_{-1} \right) - \nu _{s}\left( \frac{\partial }{\partial r} + \frac{1}{r}\right) \hat{v}_{-1} - \nu _s\frac{\mathrm {i}m}{r} \hat{w}_{-1} \right]=0,\end{split}\) 
